Báo cáo hóa học: "USING SUPERMODELS IN QUANTUM OPTICS NICOLE GARBERS AND ANDREAS RUFFING "

USING SUPERMODELS IN QUANTUM OPTICS

NICOLE GARBERS AND ANDREAS RUFFING

Received 27 January 2006; Revised 11 April 2006; Accepted 12 April 2006

Starting from supersymmetric quantum mechanics and related supermodels within Schr¨odinger theory, we review the meaning of self-similar superpotentials which exhibit the spectrum of a geometric series. We construct special types of discretizations of the Schr¨odinger equation on time scales with particular symmetries. This discretization leads to the same type of point spectrum for the referred Schr¨odinger diﬀerence operator than in the self-similar superpotential case, hence exploiting an isospectrality situation. A dis- cussion is opened on the question of how the considered energy sequence and its gener- alizations serve the understanding of coherent states in quantum optics.

Copyright © 2006 N. Garbers and A. Ruﬃng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Items like “coherent states” or “squeezed states” can nowadays be found in many recent articles on quantum optics. The fact that the Nobel Prize in Physics 2005 has been awarded to pioneers on this area, like R. Glauber, gives insight how active this area is.

The kind of physical states behind coherent states or squeezed states are the so-called nonclassical states. They are minimal uncertainty states. These properties are essential for an eﬃcient signal transmission in the quantum world. The theory of coherent states in physics has been developed all over the last decades, among others by Glauber, Klauder, and Sudarshan.

Hindawi Publishing Corporation Advances in Diﬀerence Equations Volume 2006, Article ID 72768, Pages 1–14 DOI 10.1155/ADE/2006/72768

Coherent states play a major role in laser physics. The mathematical modeling in laser physics allows three diﬀerent approaches to coherent states: ﬁrst by the method of trans- lation operators, second by the method of ladder operators, and third by the method of minimal uncertainty. Nonclassical states like squeezed laser ﬁelds are very important for applications: the experimental methods when dealing with squeezed laser ﬁelds include for instance the so-called self-homodyne tomography. The mathematical modeling in self- homodyne tomography allows a tomographical reconstruction of the Wigner function, belonging to a set of probability densities of ﬂuctuations in diﬀerent ﬁeld amplitudes.

2 Using supermodels in quantum optics

Squeezed laser states are—like coherent states—states of minimal uncertainty. From the viewpoint of statistics, semiconductor lasers have a super-Poisson distribution up to the pumping level, that is, a distribution which goes beyond the non-normalized Pois- son distribution. But also the so-called sub-Poisson distribution has a particular mean- ing in multiboson systems: the deﬁnition of coherent states is directly related to solu- tions of Stieltjes moment problems. In [4], Penson and Solomon could show that q- discretizations of orthogonality measures, solving the moment problems, allow to in- vestigate multiboson coherent states which could not so far be understood in the con- ventional quantum setting. The nature of these states is sub-Poissonian. Regarding the statistical properties of diﬀerent physical quantum states in quantum optics, one also refers to super-Poisson states and sub-Poisson states.

Apart from the development in quantum optics, there are also great achievements visi- ble in mathematical physics which support the theoretical understanding of the described phenomena. On the one hand, the mathematical frame for coherent states is steadily de- veloped throughout analysis, on the other hand one can see the development that super- symmetric quantum mechanics and discrete Schr¨odinger theory become related to essential problems in quantum optics.

In this article, we are going to exploit the connections between self-similar supermodels within supersymmetric quantum mechanics, so-called basic versions of coherent states, and their relations to discretized Schr¨odinger equations on time scales. In Sections 2 and 3, we review some fundamental facts on supersymmetric Schr¨odinger operators where we are going to exploit a generalized supermodel deﬁnition at the end of Section 4. We are going to represent generalized supermodels through their creation and annihilation op- erators and characterize them as solutions to a new type of discretization for Schr¨odinger operators in Section 4. Finally, in Section 5, we are going to establish the connection of the obtained results and representations to basic items of coherent state theory in quan- tum optics.

2. Schr¨odinger equations and superpotentials

Supersymmetry is one of the most powerful tools being applied to problems of theoretical physics. In the last years, there were great achievements especially on the area of super- symmetry in quantum mechanics. For an excellent contribution to the topic see for in- stance the articles by Robnik [7] and Robnik and Liu [8]. The stationary one-dimensional version of Schr¨odinger’s equation (λ being a ﬁxed value in R) is

(2.1)

−ψ (cid:2)(cid:2)(x) + V (x)ψ(x) = λψ(x),

x ∈ R.

It can with some general success be factorized by using the concept of so-called superpo- tentials.

In Schr¨odinger theory, the following scenario is of particular interest. Given two Schr¨odinger equations with diﬀerent potentials V1 and V2. Under some circumstances, it is possible to write them in the form

(2.2) BB+ψ = μψ, B+Bϕ = λϕ,

N. Garbers and A. Ruﬃng 3

where B and B+ are formally adjoint to each other, being deﬁned on some common do- main in (cid:2)2(R). Let us shortly review the method of how to address the stated factoriza- tion problem. The ﬁrst step is the construction of a so-called superpotential W such that one can

√

(cid:3) ,

(2.3)

(cid:2) W 2 −

2W (cid:2)

(cid:2) W 2 +

(cid:3) .

2W (cid:2) express the partner potentials V1, V2 as follows: V1 = 1 2 V2 = 1 2

The superpotential W is ﬁxed by assuming that the potential V1 allows 0 as an eigenvalue, the corresponding eigenfunction ϕ being positive. This leads to the condition

√

(2.4)

−ϕ(cid:2)(cid:2)(x) +

(cid:2) W 2(x) −

2W (cid:2)(x)

(cid:3) ϕ(x) = 0,

x ∈ R. 1 2

A solution to this equation is given by

√

(2.5) W(x) = − 2(lnϕ)(cid:2)(x), x ∈ R.

The aimed factorization is now achieved by the equalities

(2.6) H1 = B+B, H2 = BB+,

where the diﬀerential operators H1, H2 are speciﬁed by the supersymmetric ladder opera- tors

(cid:4)

(cid:5)

(cid:4)

(cid:5)

√ 2

(2.7) W + W − . B := 1√ 2 B+ := 1√ 2 d dx d dx

To illustrate this formalism, let us consider the two potentials

(2.8) , + x ∈ R. V1(x) = x2 4

− 1 2

, V2(x) = x2 4 1 2

√

2x, leading to the well-understood As for the superpotential W, we obtain just W(x) = conventional ladder operator formalism

(cid:4)

(2.9) H1 = B+B, (cid:5) √ , 2

√ 2

x + x −

(cid:5) .

B = 1√ 2 H2 = BB+, (cid:4) B+ = 1√ 2 d dx d dx

The key message is now that one can determine the point spectrum of H1, H2 completely by using the operators B, B+. Further, more illustrative example is the so-called Rosen- Morse potential, in which a real parameter y occurs:

(2.10) , x ∈ R. V1(x, y) = y2 − y(y + 1) cosh2(x)

Assuming that the equation

(2.11)

−ϕ(cid:2)(cid:2)(x) + V1(x, y)ϕ(x) = 0

4 Using supermodels in quantum optics

has a solution ϕ ∈ (cid:2)2(R), we are ﬁrst led to the superpotential

√

(2.12) W(x) = 2y tanh(x), x ∈ R,

as well as to the potential

, (2.13) x ∈ R. V2(x, y) = y2 − y(y − 1) cosh2(x)

(2.14) , x ∈ R. The diﬀerence of the two partner potentials is given by V2(x, y) − V1(x, y − 1) = 2y − 1 2

Generalizing the observations made so far, the two potentials V1, V2 are called form- invariant if the following identity holds for diﬀerent values of x, y1, y2, the expression R(y1) being a continuous function of y1:

(cid:3)

(cid:2)

(2.15) + R

(cid:3) .

(cid:2) x, y1

(cid:2) x, y2

= V1

According to the above construction, the pair B, B+ speciﬁes two diﬀerent Schr¨odinger equations, which together are referred to as a supersymmetric model or just as a super- model.

3. Self-similar supermodels

In many important applications, it follows from the deﬁning equation for form invari- ance,

(cid:3)

(cid:2)

(cid:3) ,

(3.1) + R

(cid:2) x, y1

(cid:2) x, y2

= V1

that y2 = y1 + h, where h is a real constant. A completely new class of form-invariant potentials has been proposed in [3], where potentials were constructed whose parameters are related to each other by

(3.2) 0 < q < 1. y2 = qy1,

∞(cid:6)

In order to make apparent what kind of possible point spectrum is generated by the prop- erty (3.2), we follow the basic outline in [2], where the superpotential is expanded as follows:

j=0

(3.3) W(x, y) = g j(x)y j

∞(cid:6)

for some suitable parameter, y ∈ R. The function R from (3.3) is assumed to be given by an analytic ansatz

j=0

(3.4) R(y) = R j y j.

N. Garbers and A. Ruﬃng 5

Plugging this ansatz into formula (3.1) yields

R(y) = V2(x, y) − V1(x, qy)

(cid:2)(cid:2)

(3.5)

√ 2

= W 2(x, y) − W 2(x, qy) +

(cid:3) (x, y) +

(cid:3) (x, qy)

(cid:3) .

(cid:2) ∂xW

∂xW

∞(cid:6)

Inserting now the expansion (3.4) for the function R, one obtains, by comparing the coeﬃcients,

(cid:3)(cid:2)

(3.6)

(cid:2) 1 + qn−i

1 − qi

√ (cid:2) 1 + qn 2

n ∈ N,

(cid:3) gign−i +

(cid:3) g (cid:2) n,

i=1

Rn = 1 2

0. With the abbreviations

and the value R0 being given by R0 = g (cid:2)

(3.7) n ∈ N, rn := Rn 1 − qn , dn := 1 − qn 1 + qn ,

one is led to nonlinear integral equations, given by

(cid:8)

(cid:9)

(cid:7)

n−1(cid:6)

i=1

(3.8) dt, x ∈ R, n ∈ N, 2rn − gi(t)gn−i(t) gn(x) = dn√ 2

where restrictions of the solutions of these equations are put by the conditions

(3.9) n ∈ N, R0 = 0, g0(x) = 0, rn = zδn1,

δn1 denoting the Kronecker symbol and z being a positive parameter. This nonlinear in- tegral equation allows now the solutions

(3.10) x ∈ R, n ∈ N, R(y) = R1 y = R, βnx2n−1, gn(x) = 1√ 2

n−1(cid:6)

where the coeﬃcients βn are ﬁxed by the recurrence formula

i=1

(3.11) , n ∈ N \ {1}. β0 = 0, βiβn−i, β1 = 2R1 1 + q βn = − dn 2n − 1

The superpotential now reads

(cid:4)

(cid:5)2 j−1

∞(cid:6)

j=1

, (3.12) W(x, y) = x ∈ R. β j y j x√ 2

(cid:10)∞

j=1(β j /2 j)y j (x/

√ 2)2 j ,

The formal ground-state, belonging to V1, is given by the formula

(3.13) x ∈ R. ψ0(x, y) = Ce−

(cid:3)

Direct calculation leads to the formula (cid:11)

(cid:2)(cid:11)

=

(cid:3) ,

(3.14) qW x ∈ R, W

(cid:2) x, y2

qx, y1

6 Using supermodels in quantum optics

n−1(cid:6)

showing some self-similar property. This type of superpotential is therefore also referred to by the name self-similar superpotential. Applying a generalized version of the ladder operator formalism, one is led to the energies of the operator H1, being given by

j=0

(3.15) , q j = R n ∈ N0. λn = R 1 − qn 1 − q

Let us now arrive at an interesting isospectrality scenario.

4. Isospectral supermodels and strip discretizations

We now address the general question of how to construct discrete Schr¨odinger oper- ators, that is, Schr¨odinger diﬀerence operators whose wave functions are deﬁned on a nonempty closed set Ω ⊂ R with Lebesgue measure μ(Ω) > 0, leading to the same point spectrum (3.15).

We address this question in context of Schr¨odinger q-diﬀerence equations, where we study piecewise continuous solutions to these equations, having support on some kind of strip structures which are generated by the symmetries of the lattice {+qn, −qn | n ∈ Z}. This approach seems to ﬁt naturally to the given point spectrum (3.15) and might be of importance for applications and numerical investigations of the underlying eigenvalue and spectral problems.

Let us now elucidate in some detail the philosophy of using the framework of q- diﬀerence operators for discretizing the Schr¨odinger equation. As indicated, we restrict our investigations to subsets of the real axis which we will call homogeneous q-strip dis- cretizations or just strip discretizations. To do so, we have to provide the tools that help us in formulating the special boundary conditions. Let us refer throughout the sequel to a parameter 0 < q < 1, as it was motivated by the

investigation of self-similar superpotentials in the previous section. Deﬁnition 4.1 (strip discretization). Let Ω ⊆ R \ {0} be a nonempty closed set with Lebesgue measure μ(Ω) > 0 as well as

(4.1)

∀x ∈ Ω,

qx ∈ Ω, q−1x ∈ Ω, −x ∈ Ω.

We call the time scale Ω a homogeneous strip discretization or just strip discretization of the conﬁguration space. The Hilbert space of the strip discretization is introduced by the requirement

(cid:12)

(cid:2)2(Ω) :=

(cid:13) ,

(4.2) f ∈ (cid:2)2(R)| f = f ◦ χΩ

and the scalar product of two functions f ,g ∈ (cid:2)2(Ω) is introduced by

(cid:7) ∞

−∞

(4.3) f (x)g(x)χΩ(x)dx = f (x)g(x)dx, ( f ,g)Ω :=

using the characteristic function χΩ of the time scale Ω. By construction, it is clear that (cid:2)2(Ω) is a Hilbert space over C, being a proper subspace of the square-integrable func- tions themselves, that is, of (cid:2)2(R). In order to proceed, let us ﬁrst review some facts on

N. Garbers and A. Ruﬃng 7

the Schr¨odinger equation with quadratic potential, given by

(4.4)

−ψ (cid:2)(cid:2)(x) + x2ψ(x) = λψ(x),

x ∈ R.

The following structure is one the most familiar facts within mathematical physics: let the sequence of functions (ψn)n∈N0 be recursively given by the requirement

n(x) + xψn(x),

(4.5) x ∈ R, n ∈ N0, ψn+1(x) := −ψ (cid:2)

where ψ0 : R → R, x (cid:11)→ ψ0(x) := e−(1/2)x2. We then have ψn ∈ (cid:2)2(R) ∩ C2(R) for n ∈ N0 and moreover

(4.6)

−ψ (cid:2)(cid:2)

n (x) + x2ψn(x) = (2n + 1)ψn(x),

x ∈ R, n ∈ N0.

This result reﬂects the conventional ladder operator formalism. We now develop a result in discrete Schr¨odinger theory on strip structures which turns out to be a q-analog of the just described continuous situation. Let us therefore state in a next step some more useful tools for the strip discretization approach. Deﬁnition 4.2. Let Ω ⊆ R \ {0} be a nonempty closed set with the property μ(Ω) > 0 as well as

(4.7)

∀x ∈ Ω,

qx ∈ Ω, q−1x ∈ Ω, −x ∈ Ω.

Let for any f : Ω → R the right-shift, respectively, left-shift operations be deﬁned by

(cid:2)

(cid:3) ,

(4.8) (R f )(x) := f (qx), (L f )(x) := f q−1x x ∈ Ω,

respectively. The right-hand, respectively, left-hand q-diﬀerence operations will for any f : Ω → R be given by

(cid:2)

(cid:3)

(cid:2)

(cid:3)

− f (x)

(4.9) , , (x) := f (qx) − f (x)

(cid:3) (x) := f

x ∈ Ω. Dq f

(cid:2) Dq−1 f

qx − x q−1x q−1x − x

Let moreover α > 0 and let

(cid:14)

ϕ(x) ϕ(qx) − (cid:14)

=

, (4.10) x (cid:11)−→ g(x) := g : Ω −→ R+, 1 + α(1 − q)x2 − 1 qx − x ϕ(x)(q − 1)x

where the positive even continuous function ϕ : Ω → R+ is chosen as a solution to the q-diﬀerence equation

(4.11)

(cid:2) 1 + α(1 − q)x2

(cid:3) ϕ(x),

x ∈ Ω. ϕ(qx) =

We are now able to deﬁne discrete ladder operators on strip structures.

8 Using supermodels in quantum optics

q and, respectively, annihilation operation Aq are introduced

The creation operation A†

by their actions on any ψ : Ω → R as follows: (cid:2) (4.12)

(cid:3) ψ,

(cid:3) ψ.

A†

− Dq + g(X)R

Aqψ = q−1

(cid:2) LDq + Lg(X)

q ψ =

We refer to the discrete Schr¨odinger equation with an oscillator potential on Ω by

(cid:2)

(cid:3)(cid:2)

(4.13)

(cid:3) ψ = λψ.

q−1

− Dq + g(X)R

LDq + Lg(X)

The following result reveals that the discrete Schr¨odinger equation with an oscillator

potential on Ω shows similar properties than its classical analog does. Theorem 4.3. Let the time scale Ω be a strip discretization in the sense of Deﬁnition 4.1 and let the function ϕ be speciﬁed like in Deﬁnition 4.2, satisfying the q-diﬀerence equation (4.11) on Ω,

(cid:2)

(4.14) 1 + α(1 − q)x2

(cid:3) ϕ(x), ϕ(x) = ϕ(−x) > 0,

x ∈ Ω. ϕ(qx) =

√

q )n

ϕ)(x), x ∈ Ω, are well For n ∈ N0, the functions ψn : Ω → R, given by ψn(x) := ((A† deﬁned in (cid:2)2(Ω) and solve the q-Schr¨odinger equation (4.13) in the following sense:

(cid:2)

(cid:3)(cid:2)

(4.15) q−1

− Dq + g(X)R

LDq + Lg(X) ψn.

(cid:3) ψn = α q

q act as ladder operators on the functions (ψn)n∈N0 in the

qn − 1 q − 1

Moreover, the linear maps Aq, A† following sense (n ∈ N0, ψ−1 := 0):

n (x)ψ0(x),

q ψn = ψn+1,

n : Ω → R are given by

(4.16) x ∈ Ω, A† ψn−1, ψn(x) = H q Aqψn = α q qn − 1 q − 1

where for n ∈ N0, the functions H q

n (x) + α

0 (x) = 1,

1 (x) = αx.

n+1(x) − αqnxH q

n−1(x) = 0,

−1 := 0 is set.

H q H q H q H q qn − 1 q − 1 (4.17)

(cid:3)

(4.18)

(cid:2) A†

Ω, m,n ∈ N0,

These recurrence relations apply for x ∈ Ω and n ∈ N0, where ψ−1 := 0, H q There exists the general observation (cid:3) Ω =

(cid:2) ψm,Aqψn

q ψm,ψn

and the functions (ψn)n∈N0 constitute an orthonormal system in (cid:2)2(Ω). Proof. Let us for ϕ ∈ C(R) ﬁrst consider the equation

(cid:2)

(4.19) 1 + α(1 − q)x2

(cid:3) ϕ(x)xn,

ϕ(qx)xn = x ∈ Ω, n ∈ N0,

which directly follows from (4.11). Using standard arguments, one can show that the

N. Garbers and A. Ruﬃng 9

function ϕ fulﬁling (4.11) is in (cid:2)1(R). This implies

(cid:7) ∞

(cid:3)

(4.20) ϕ(qx)xnχΩ(x)dx = ϕ(x)xnχΩ(x)dx,

(cid:2) 1 + α(1 − q)x2

−∞

n ∈ N0.

Using the substitution rule to the left-hand side, this directly implies

(cid:7) ∞

(cid:2)

ϕ(t)tnq−nχΩ(q−1t)q−1dt = 1 + α(1 − q)x2

(cid:3) ϕ(x)xnχΩ(x)dx,

−∞

n ∈ N0.

(4.21)

Because of (4.7) we have χΩ(q−1t) = χΩ(t) for any t ∈ R and, therefore, (4.21) is equiva- lent to (cid:7) ∞

(cid:7) ∞

(4.22) ϕ(t)tnq−nχΩ(t)q−1dt =

(cid:2) 1 + α(1 − q)x2

(cid:3) ϕ(x)xnχΩ(x)dx,

−∞

−∞ (cid:15) Ω xnϕ(x)dx for n ∈ N0 we obtain the following result:

n ∈ N0.

Using the abbreviation μn(Ω) :=

(4.23) n ∈ N0. μ2n(Ω), μ2n+1(Ω) = 0, μ2n+2(Ω) = q−2n−1 − 1 α(1 − q)

We have shown earlier [1] that any probability density ψ which generates moments of type (4.23) yields an orthogonality measure to the polynomials (H q n )n∈N0 which are for k ∈ N ﬁxed through the recurrence relation

0 (x) = 1,

1 (x) = αx,

k+1(x) − αqkxH q H q

k (x) + α

k−1(x) = 0,

H q H q H q qk − 1 q − 1 (4.24)

the variable x being chosen in a suitable integration support. As a consequence of this general result, we obtain the following orthogonality relation:

(cid:7) ∞

m(x)H q

n (x)ϕ(x)χΩ(x)dx = vn(Ω)δmn, m,n ∈ N0,

−∞

(4.25) H q

with a sequence of positive numbers (vn(Ω))n∈N0. Direct calculations and induction show

(cid:2)(cid:2)

(cid:3)

(cid:11)

(cid:3) n(cid:11)

(4.26) ϕ ◦ χΩ (x) =

(cid:2) H q

ϕ ◦ χΩ

(cid:3) (x),

n (X)

x ∈ R, n ∈ N0. ψn(x) := A† q

Let us from now on—without any restriction—refer to the special parameter choice α = 1. The functions (ψn)n∈N0 constitute an orthonormal system in (cid:2)2(Ω). Let us show next that the ladder property (4.16) is fulﬁled. The ﬁrst equation in (4.16) is trivial due to the deﬁnition of the functions (ψn)n∈N0. We remember that the function g is speciﬁed like in Deﬁnition 4.2. We obtain in the sense of the multiplication operator notation

(cid:3)(cid:2)

(cid:3)

(4.27) X nψ0 n ∈ N0,

(cid:2) LDq + Lg(X)

= LDqX nψ0 + Lg(X)X nψ0,

10 Using supermodels in quantum optics

which yields

(cid:4)

(cid:5)

(cid:3)(cid:2)

(cid:3)

= L

X nψ0 + Lg(X)X nψ0, n ∈ N0.

(cid:2) LDq + Lg(X)

X n−1Rψ0 + X nDqψ0 qn − 1 q − 1 (4.28)

This may be rewritten as

(cid:3)(cid:2)

(cid:3)

(cid:3) ,

(4.29) X nψ0 q−n+1X n−1ψ0 + LX n n ∈ N0.

(cid:2) LDq + Lg(X)

(cid:2) Dqψ0 + gψ0

= qn − 1 q − 1

Using now however the formulas in (4.10) for the function g, we obtain (Dqψ0 + gψ0) = 0 and therefore

(cid:3)(cid:2)

(cid:3)

(4.30) n ∈ N. X nψ0 q−n+1X n−1ψ0,

(cid:2) LDq + Lg(X)

= qn − 1 q − 1

n−1(cid:6)

For m ∈ N0, the ﬁrst m + 1 polynomials of the sequence (H q n (X))n∈N0 can uniquely be generated by linear combinations of the ﬁrst m + 1 monomials of the sequence (X n)n∈N0. We therefore conclude as

j=0

(4.31) n ∈ N, Aqψn = cn j ψ j,

with uniquely deﬁned real numbers cn j , where j = 0,...,n − 1 with n ∈ N. Applying again standard substitution techniques to the scalar product integral (4.3), we can derive for any functions f ,g ∈ (cid:2)2(Ω) which are both in the algebraic span of the functions (ψn)n∈N0 the following relation:

(cid:2)

(4.32)

(cid:2) A†

(cid:3) Ω =

(cid:3) Ω.

q f ,g

f ,Aqg

In particular, this result implies

(cid:2)

(4.33)

(cid:2) A†

(cid:3) Ω =

(cid:3) Ω, m,n ∈ N0.

q ψm,ψn

ψm,Aqψn

Using the ﬁrst equation in (4.16) and because of the fact that the functions (ψn)n∈N0 constitute an orthogonal system in (cid:2)2(Ω), the second relation in (4.16) follows from standard methods of calculating the norms of the functions (ψn)n∈N0. Equation (4.15) now follows immediately from the ﬁrst two relations in (4.16). Taking all the steps of the proof together, this ﬁnally conﬁrms the statements of Theorem 4.3.

Let us interpret the obtained results in context of quantum mechanical supermodels. First, we have obtained the desired isospectrality result, that is, we have found a rich class of discrete Schr¨odinger operators showing in (4.15) the same point spectrum that we obtain from the self-similar superpotentials in (3.15). This gives us an important tool at hand to extend the deﬁnition of a supermodel for purposes of quantum optics.

As the discrete ladder operator formalism that we have revealed and orthogonal eigen- systems for self-similar superpotentials lead to the same point spectrum, ﬁxed by (4.15), both type of solutions may be considered as two diﬀerent representations of one and the (cid:2) same formal orthogonal system.

N. Garbers and A. Ruﬃng 11

This is the reason why we introduce the following general deﬁnition for a self-similar supermodel.

(4.34) , n ∈ N0, Deﬁnition 4.4. Let for 0 < q < 1 the bounded sequence λ = (λn)n∈N0 of eigenvalues λn := qn − 1 q − 1

be given, and let moreover the subspace l2(λ) ⊆ l2(N0) be canonically introduced as fol- lows:

(cid:18)

(cid:16)

∞(cid:6)

(cid:3)

(cid:17) (cid:17)2

|

(4.35) ψ = < ∞ .

(cid:2) N0

cnen ∈ l2

(cid:17) (cid:17)cn

n=0

n=1

λnλn−1

(cid:3)

(cid:3) ,

(4.36) Let the pair of adjoint operators (cid:2) A†

⊆ l2(λ) −→ l2

A† : Dmax

(cid:2) N0

A : Dmax(A) ⊆ l2(λ) −→ l2

(cid:2) N0

be given, being ﬁxed by their actions on the standard orthogonal basis vectors of l2(λ) ⊆ l2(N0) as follows:

(cid:14)

(4.37) n ∈ N0. A†en := λn+1en+1, Aen := λnen,

Then the triple (A,A†,l2(λ)) is called self-similar supermodel in Fock space. As a direct consequence, we see that A and A† fulﬁl on a common maximum domain of l2(λ) ⊆ l2(N0) the commutation relation

(4.38) AA† − qA†A = E,

the operator E denoting the identity map on the maximal common domain of A, A†.

By Deﬁnition 4.4, we give an abstract formulation of the ladder operator formalism for which the self-similar supermodels from Section 3 and the discrete Schr¨odinger models from Section 4 so far are two diﬀerent realizations. Therefore, an eigenvalue distribution of type (4.34) has indeed a physical interpretation in Schr¨odinger theory. We now may ask what is the impact of this type of spectrum for coherent state theory within quantum optics. A discussion on this topic will be started by the next section.

5. Discussion of applications to coherent state theory

In the introduction, we have mentioned the role of super-Poisson states respectively, sub-Poisson states in quantum optics. A mathematical model for distinguishing between super-Poisson states and sub-Poisson states is given by the Mandel functional, see for instance [5, 6]. The characterization of Mandel’s functional is associated with Jacobi op- erators in l2(N0). The physical states of the quantum system are modelled by the elements of l2(N0). (cid:10)∞

(cid:10)∞

n=0 cnen ∈ l2(N0) with the property

ψ = One ﬁrst considers the C-linear subspace l1 ⊆ l2(N0) which is described by all elements n=1 n|cn|2 < ∞. Let in this context (en)n∈N0 be

12 Using supermodels in quantum optics

a standard orthonormal basis of l2(N0). Moreover, let (cid:3) (5.1) l2 ⊆ l1 ⊆ l2

(cid:2) N0

denote the particular C-linear subspace which is deﬁned by all the elements ψ = (cid:10)∞

(cid:10)∞

n=0 cnen ∈ l2(N0) with the property

n=1 n(n − 1)|cn|2 < ∞. For an arbitrary element

∞(cid:6)

n=1

(5.2) ψ = cnen ∈ l2,

∞(cid:6)

the nonlinear Mandel functional f : l2 → R is deﬁned by

(cid:17) (cid:17)2

(cid:17) (cid:17)2 −

(5.3) f (ψ) := n(n − 1) n .

(cid:17) (cid:17)cn

(cid:10)∞

(cid:17) (cid:17)2

(cid:17) (cid:17)cn

n=1

1 n=1 n

Mandel’s functional now allows to distinguish between super-Poisson states and sub- Poisson states of the quantum system. On super-Poisson states ψ ∈ l2, we have f (ψ) > 0, and on sub-Poisson states ψ ∈ l2, we have f (ψ) < 0. States which are associated with the Poisson distribution itself, are characterized by a vanishing Mandel functional. Let now U ⊆ C be a connected set and let

(cid:2)

(cid:3)

(cid:13)

(5.4)

| z ∈ U

N0

ΨU :=

(cid:12) ψz ∈ l2

ΨU is modeling quantum states on which a continuous label z from U has been ﬁxed. The elements of ΨU are referred to by the name coherent states if

(cid:19) (cid:19)

(5.5)

(cid:3) −→ 0

zn −→ z =⇒

− ψz

(cid:19) (cid:19)ψzn

(cid:2) N0

for any convergent sequence (zn)n∈N in U and if they allow a decomposition of the iden- tity map in the following sense:

(cid:7)

(cid:3)

(5.6) :

∀ϕ ∈ l2

(cid:2) |z|2

(cid:2) N0

(cid:3) ψ ∗ z (ϕ)ψzdz = ϕ.

→ R+ is a suitable weight function and ψ ∗ In this context, w : R+ z denotes a canonically 0 constructed dual form on l2(N0), its construction being inherited from the structure of ΨU . Physical states which arise in the simplest quantum systems, for instance, are mod- eled by

∞(cid:6)

n=0

(5.7) z ∈ U = C. ψz = e−(1/2)|z|2 en, zn n!

In the sense of the above-given deﬁnition, these functions indeed are coherent states. They are states which are eigenvectors of the linear map being ﬁxed by

(cid:3)

√

(cid:3) ,

(5.8)

−→ l2

n ∈ N, A : Dmax(A) ⊆ l2

(cid:2) N0

Aen := nen−1,

N. Garbers and A. Ruﬃng 13

on the maximal possible deﬁnition range for the C-linear map A. The map A is modeling the transition from a particular energy level of the quantum system to its lower neigh- bor. In this way, the coherent states which are ﬁxed by (4.26) are stable towards energy perturbations. This is one of their most important properties, being fundamental for the respective applications in quantum optics. This stability is mathematically modeled by the following eigenvalue equation:

(5.9) z ∈ C. Aϕz = zϕz,

The deﬁnition of a so-called squeezed state is given by the variance vQ(ψ) of a linear op- erator Q in ψ ∈ l2(N0). Let Q and Q2 be deﬁned on a common dense domain Δ ⊆ l2(N0). The nonlinear variance functional vQ : Δ → C is expressed in terms of the canonical scalar product as follows:

(cid:3)

(5.10)

(cid:2) ψ,Q2ψ

− (ψ,Qψ)2, ψ ∈ Δ.

vQ(ψ) :=

Let now A∗ be the adjoint of the operator A in (5.8). Let moreover the following operators be given, sharing the joint deﬁnition ranges of A, respectively, A∗ in l2(N0):

(cid:3) ,

(5.11) P :=

(cid:2) A − A∗

(cid:2) A + A∗

(cid:3) .

−i√ 2

X := 1√ 2

One has then for all elements ψ of a common deﬁnition range Δ:

(5.12) , ψ ∈ Δ. Heisenberg uncertainty relation vP(ψ)vX (ψ) ≥ 1 4

In this situation, one calls

(5.13) . ψ ∈ Δ squeezed state: ⇐⇒ vX (ψ) < 1 2

Thus it is a state which has only a small freedom of variation in the spatial sense.

∞(cid:6)

Starting from a suitable sequence λ = (λn)n∈N0 of positive numbers modeling the en- ergies of a given quantum system, on can investigate generalizations of the Mandel func- tional fλ : l2(λ) → R and study their properties:

(cid:17) (cid:17)2 −

(5.14)

(cid:17) (cid:17)2, ψ ∈ l2(λ).

fλ(ψ) :=

(cid:17) (cid:17)cn

λnλn−1

(cid:17) (cid:17)cn

λn

(cid:10)∞

(cid:17) (cid:17)2

(cid:17) (cid:17)cn

n=1

1 n=1 λn

The subspace l2(λ) ⊆ l2(N0) is canonically given as a generalization of the situation (5.1),

(cid:18)

(cid:16)

∞(cid:6)

(cid:3)

(cid:17) (cid:17)2

|

(5.15) ψ = < ∞ .

(cid:2) N0

cnen ∈ l2

(cid:17) (cid:17)cn

n=0

n=1

λnλn−1

The following question is of particular interest, given a special energy sequence (λn)n∈N0. How often do the super-Poisson distribution, respectively, the sub-Poisson distribution oc- cur? Discussing this question for the energy sequence of the self-similar supermodel will therefore be the detailed subject of a diﬀerent article.

14 Using supermodels in quantum optics

[1] K. Ey and A. Ruﬃng, The moment problem of discrete q-Hermite polynomials on diﬀerent time

scales, Dynamic Systems and Applications 13 (2004), no. 3-4, 409–418.

[2] H. Kalka and G. Soﬀ, Supersymmetrie, Teubner, Stuttgart, 1997. [3] A. Khare and U. P. Sukhatme, New shape-invariant potentials in supersymmetric quantum me-

chanics, Journal of Physics. A 26 (1993), no. 18, L901–L904.

[4] K. A. Penson and A. I. Solomon, New generalized coherent states, Journal of Mathematical Physics

References

40 (1999), no. 5, 2354–2363.

[5] C. Quesne, K. A. Penson, and V. M. Tkachuk, Maths-type q-deformed coherent states for q > 1,

Physics Letters. A 313 (2003), no. 1-2, 29–36.

[6]

, Geometrical and physical properties of maths-type q-deformed coherent states for 0 < q < 1

and q > 1, preprint, 2003.

[7] M. Robnik, Supersymmetric quantum mechanics based on higher excited states, Journal of Physics.

A 30 (1997), no. 4, 1287–1294.

[8] M. Robnik and J. Liu, Supersymmetric quantum Mechanics based on higher excited states II: a few

new examples of isospectral partner potentials, preprint CAMTP, 1997.

Nicole Garbers: Fakult¨at f¨ur Mathematik, Technische Universit¨at M¨unchen, Boltzmannstraße 3, Garching 85747, Germany E-mail address: ngarbers@web.de

Andreas Ruﬃng: Fakult¨at f¨ur Mathematik, Technische Universit¨at M¨unchen, Boltzmannstraße 3, Garching 85747, Germany E-mail address: ruﬃng@ma.tum.de